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Real exponential sums over primes and prime gaps (2307.08725v1)
Published 17 Jul 2023 in math.NT
Abstract: We prove that given $\lambda \in \mathbb{R}$ such that $0 < \lambda < 1$, then $\pi(x + x\lambda) - \pi(x) \sim \displaystyle \frac{x\lambda}{\log(x)}$. This solves a long-standing problem concerning the existence of primes in short intervals. In particular, we give a positive answer (for all sufficiently large number) to some old conjectures about prime numbers, such as Legendre's conjecture about the existence of at least two primes between two consecutive squares.